Delayed population models with Allee effects and exploitation

Previous Article
Dynamics of competitive systems with a single common limiting factor
MBE Home
This Issue
Next Article
Global dynamics of a general class of multi-group epidemic models with latency and relapse

2015, 12(1): 83-97. doi: 10.3934/mbe.2015.12.83

Delayed population models with Allee effects and exploitation

Eduardo Liz ^1, and Alfonso Ruiz-Herrera ^2,

1.	Departamento de Matemática Aplicada II, E.T.S.E. Telecomunicación, Universidade de Vigo, Campus Marcosende, 36310 Vigo
2.	Bolyai Institute, University of Szeged, Aradi vértanúk tere 1., H-6720 Szeged, Hungary

Received April 2014 Revised October 2014 Published December 2014

Abstract
Related Papers

Allee effects make populations more vulnerable to extinction, especially under severe harvesting or predation. Using a delay-differential equation modeling the evolution of a single-species population subject to constant effort harvesting, we show that the interplay between harvest strength and Allee effects leads not only to collapses due to overexploitation; large delays can interact with Allee effects to produce extinction at population densities that would survive for smaller time delays. In case of bistability, our estimations on the basins of attraction of the two coexisting attractors improve some recent results in this direction. Moreover, we show that the persistent attractor can exhibit bubbling: a stable equilibrium loses its stability as harvesting effort increases, giving rise to sustained oscillations, but higher mortality rates stabilize the equilibrium again.

Keywords: bifurcation., bistability, difference equation, global stability, delay differential equation, Allee effect, Population model.

Mathematics Subject Classification: Primary: 34K18, 34K20, 92D25; Secondary: 37E0.

Citation: Eduardo Liz, Alfonso Ruiz-Herrera. Delayed population models with Allee effects and exploitation. Mathematical Biosciences & Engineering, 2015, 12 (1) : 83-97. doi: 10.3934/mbe.2015.12.83

References:

[1]	D. S. Boukal and L. Berec, Single-species models of the Allee effect: Extinction boundaries, sex ratios and mate encounters,, J. Theoret. Biol., 218 (2002), 375. doi: 10.1006/jtbi.2002.3084.
[2]	B. Cid, F. M. Hilker and E. Liz, Harvest timing and its population dynamic consequences in a discrete single-species model,, Math. Biosci., 248 (2014), 78. doi: 10.1016/j.mbs.2013.12.003.
[3]	C. W. Clark, Mathematical Bioeconomics. Optimal Management of Renewable Resources,, $2^{nd}$ edition, (1990).
[4]	F. Courchamp, L. Berec and J. Gascoigne, Allee Effects in Ecology and Conservation,, Oxford University Press, (2008). doi: 10.1093/acprof:oso/9780198570301.001.0001.
[5]	A. M. De Roos and L. Persson, Size-dependent life-history traits promote catastrophic collapses of top predators,, Proc. Natl. Acad. Sci. USA, 99 (2002), 12907.
[6]	O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel and H.-O. Walther, Delay Equations. Functional-, Complex-, and Nonlinear Analysis,, Springer-Verlag, (1995). doi: 10.1007/978-1-4612-4206-2.
[7]	S. N. Elaydi and R. J. Sacker, Population models with Allee effect: A new model,, J. Biol. Dyn., 4 (2010), 397. doi: 10.1080/17513750903377434.
[8]	S. A. H. Geritz and E. Kisdi, Mathematical ecology: Why mechanistic models?,, J. Math. Biol., 65 (2012), 1411. doi: 10.1007/s00285-011-0496-3.
[9]	K. P. Hadeler, Neutral delay equations from and for population dynamics,, Electron. J. Qual. Theory Differ. Equ., 11 (2008), 1.
[10]	C. Huang, Z. Yang, T. Yi and X. Zou, On the basins of attraction for a class of delay differential equations with non-monotone bistable nonlinearities,, J. Differential Equations, 256 (2014), 2101. doi: 10.1016/j.jde.2013.12.015.
[11]	A. F. Ivanov, E. Liz and S. Trofimchuk, Global stability of a class of scalar nonlinear delay differential equations,, Differential Equations Dynam. Systems, 11 (2003), 33.
[12]	A. F. Ivanov and A. N. Sharkovsky, Oscillations in singularly perturbed delay equations,, Dynam. Report. Expositions Dynam. Systems (N.S.), 1 (1992), 164.
[13]	M. Jankovic and S. Petrovskii, Are time delays always destabilizing? Revisiting the role of time delays and the Allee effect,, Theor. Ecol., 7 (2014), 335. doi: 10.1007/s12080-014-0222-z.
[14]	T. Krisztin, Global dynamics of delay differential equations,, Period. Math. Hungar., 56 (2008), 83. doi: 10.1007/s10998-008-5083-x.
[15]	T. Krisztin and E. Liz, Bubbles for a class of delay differential equations,, Qual. Theory Dyn. Syst., 10 (2011), 169. doi: 10.1007/s12346-011-0055-8.
[16]	Y. Kuang, Delay Differential Equations with Applications in Population Dynamics,, Academic Press, (1993).
[17]	E. Liz, Complex dynamics of survival and extinction in simple population models with harvesting,, Theor. Ecol., 3 (2010), 209. doi: 10.1007/s12080-009-0064-2.
[18]	E. Liz, M. Pinto, V. Tkachenko and S. Trofimchuk, A global stability criterion for a family of delayed population models,, Quart. Appl. Math., 63 (2005), 56.
[19]	E. Liz and G. Röst, On the global attractor of delay differential equations with unimodal feedback,, Discrete Contin. Dyn. Syst., 24 (2009), 1215. doi: 10.3934/dcds.2009.24.1215.
[20]	E. Liz and A. Ruiz-Herrera, Attractivity, multistability, and bifurcation in delayed Hopfield's model with non-monotonic feedback,, J. Differential Equations, 255 (2013), 4244. doi: 10.1016/j.jde.2013.08.007.
[21]	E. Liz, V. Tkachenko and S. Trofimchuk, A global stability criterion for scalar functional differential equations,, SIAM J. Math. Anal., 35 (2003), 596. doi: 10.1137/S0036141001399222.
[22]	J. Mallet-Paret and R. Nussbaum, Global continuation and asymptotic behaviour for periodic solutions of a differential-delay equation,, Ann. Mat. Pura Appl., 145 (1986), 33. doi: 10.1007/BF01790539.
[23]	G. Röst, On the global attractivity controversy for a delay model of hematopoiesis,, Appl. Math. Comput., 190 (2007), 846. doi: 10.1016/j.amc.2007.01.103.
[24]	G. Röst and J. Wu, Domain decomposition method for the global dynamics of delay differential equations with unimodal feedback,, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 463 (2007), 2655. doi: 10.1098/rspa.2007.1890.
[25]	S. Ruan, Delay differential equations in single species dynamics,, in Delay differential equations and applications, 205 (2006), 477. doi: 10.1007/1-4020-3647-7_11.
[26]	S. J. Schreiber, Chaos and population disappearances in simple ecological models,, J. Math. Biol., 42 (2001), 239. doi: 10.1007/s002850000070.
[27]	S. J. Schreiber, Allee effect, extinctions, and chaotic transients in simple population models,, Theor. Popul. Biol., 64 (2003), 201. doi: 10.1016/S0040-5809(03)00072-8.
[28]	A. N. Sharkovsky, S. F. Kolyada, A. G. Sivak and V. V. Fedorenko, Dynamics of One-Dimensional Maps,, Kluwer Academic Publishers, (1997). doi: 10.1007/978-94-015-8897-3.
[29]	A. N. Sharkovsky, Y. L. Maistrenko and E. Y. Romanenko, Difference Equations and Their Applications,, Kluwer Academic Publishers, (1993). doi: 10.1007/978-94-011-1763-0.
[30]	D. Singer, Stable orbits and bifurcation of maps of the interval,, SIAM J. Appl. Math., 35 (1978), 260. doi: 10.1137/0135020.
[31]	A.-A. Yakubu, N. Li, J. M. Conrad and M.-L. Zeeman, Constant proportion harvest policies: Dynamic implications in the Pacific halibut and Atlantic cod fisheries,, Math. Biosci., 232 (2011), 66. doi: 10.1016/j.mbs.2011.04.004.
[32]	T. Yi and X. Zou, Maps dynamics versus dynamics of associated delay reaction-diffusion equation with a Neumann condition,, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 466 (2010), 2955. doi: 10.1098/rspa.2009.0650.

show all references

References:

[1]	D. S. Boukal and L. Berec, Single-species models of the Allee effect: Extinction boundaries, sex ratios and mate encounters,, J. Theoret. Biol., 218 (2002), 375. doi: 10.1006/jtbi.2002.3084.
[2]	B. Cid, F. M. Hilker and E. Liz, Harvest timing and its population dynamic consequences in a discrete single-species model,, Math. Biosci., 248 (2014), 78. doi: 10.1016/j.mbs.2013.12.003.
[3]	C. W. Clark, Mathematical Bioeconomics. Optimal Management of Renewable Resources,, $2^{nd}$ edition, (1990).
[4]	F. Courchamp, L. Berec and J. Gascoigne, Allee Effects in Ecology and Conservation,, Oxford University Press, (2008). doi: 10.1093/acprof:oso/9780198570301.001.0001.
[5]	A. M. De Roos and L. Persson, Size-dependent life-history traits promote catastrophic collapses of top predators,, Proc. Natl. Acad. Sci. USA, 99 (2002), 12907.
[6]	O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel and H.-O. Walther, Delay Equations. Functional-, Complex-, and Nonlinear Analysis,, Springer-Verlag, (1995). doi: 10.1007/978-1-4612-4206-2.
[7]	S. N. Elaydi and R. J. Sacker, Population models with Allee effect: A new model,, J. Biol. Dyn., 4 (2010), 397. doi: 10.1080/17513750903377434.
[8]	S. A. H. Geritz and E. Kisdi, Mathematical ecology: Why mechanistic models?,, J. Math. Biol., 65 (2012), 1411. doi: 10.1007/s00285-011-0496-3.
[9]	K. P. Hadeler, Neutral delay equations from and for population dynamics,, Electron. J. Qual. Theory Differ. Equ., 11 (2008), 1.
[10]	C. Huang, Z. Yang, T. Yi and X. Zou, On the basins of attraction for a class of delay differential equations with non-monotone bistable nonlinearities,, J. Differential Equations, 256 (2014), 2101. doi: 10.1016/j.jde.2013.12.015.
[11]	A. F. Ivanov, E. Liz and S. Trofimchuk, Global stability of a class of scalar nonlinear delay differential equations,, Differential Equations Dynam. Systems, 11 (2003), 33.
[12]	A. F. Ivanov and A. N. Sharkovsky, Oscillations in singularly perturbed delay equations,, Dynam. Report. Expositions Dynam. Systems (N.S.), 1 (1992), 164.
[13]	M. Jankovic and S. Petrovskii, Are time delays always destabilizing? Revisiting the role of time delays and the Allee effect,, Theor. Ecol., 7 (2014), 335. doi: 10.1007/s12080-014-0222-z.
[14]	T. Krisztin, Global dynamics of delay differential equations,, Period. Math. Hungar., 56 (2008), 83. doi: 10.1007/s10998-008-5083-x.
[15]	T. Krisztin and E. Liz, Bubbles for a class of delay differential equations,, Qual. Theory Dyn. Syst., 10 (2011), 169. doi: 10.1007/s12346-011-0055-8.
[16]	Y. Kuang, Delay Differential Equations with Applications in Population Dynamics,, Academic Press, (1993).
[17]	E. Liz, Complex dynamics of survival and extinction in simple population models with harvesting,, Theor. Ecol., 3 (2010), 209. doi: 10.1007/s12080-009-0064-2.
[18]	E. Liz, M. Pinto, V. Tkachenko and S. Trofimchuk, A global stability criterion for a family of delayed population models,, Quart. Appl. Math., 63 (2005), 56.
[19]	E. Liz and G. Röst, On the global attractor of delay differential equations with unimodal feedback,, Discrete Contin. Dyn. Syst., 24 (2009), 1215. doi: 10.3934/dcds.2009.24.1215.
[20]	E. Liz and A. Ruiz-Herrera, Attractivity, multistability, and bifurcation in delayed Hopfield's model with non-monotonic feedback,, J. Differential Equations, 255 (2013), 4244. doi: 10.1016/j.jde.2013.08.007.
[21]	E. Liz, V. Tkachenko and S. Trofimchuk, A global stability criterion for scalar functional differential equations,, SIAM J. Math. Anal., 35 (2003), 596. doi: 10.1137/S0036141001399222.
[22]	J. Mallet-Paret and R. Nussbaum, Global continuation and asymptotic behaviour for periodic solutions of a differential-delay equation,, Ann. Mat. Pura Appl., 145 (1986), 33. doi: 10.1007/BF01790539.
[23]	G. Röst, On the global attractivity controversy for a delay model of hematopoiesis,, Appl. Math. Comput., 190 (2007), 846. doi: 10.1016/j.amc.2007.01.103.
[24]	G. Röst and J. Wu, Domain decomposition method for the global dynamics of delay differential equations with unimodal feedback,, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 463 (2007), 2655. doi: 10.1098/rspa.2007.1890.
[25]	S. Ruan, Delay differential equations in single species dynamics,, in Delay differential equations and applications, 205 (2006), 477. doi: 10.1007/1-4020-3647-7_11.
[26]	S. J. Schreiber, Chaos and population disappearances in simple ecological models,, J. Math. Biol., 42 (2001), 239. doi: 10.1007/s002850000070.
[27]	S. J. Schreiber, Allee effect, extinctions, and chaotic transients in simple population models,, Theor. Popul. Biol., 64 (2003), 201. doi: 10.1016/S0040-5809(03)00072-8.
[28]	A. N. Sharkovsky, S. F. Kolyada, A. G. Sivak and V. V. Fedorenko, Dynamics of One-Dimensional Maps,, Kluwer Academic Publishers, (1997). doi: 10.1007/978-94-015-8897-3.
[29]	A. N. Sharkovsky, Y. L. Maistrenko and E. Y. Romanenko, Difference Equations and Their Applications,, Kluwer Academic Publishers, (1993). doi: 10.1007/978-94-011-1763-0.
[30]	D. Singer, Stable orbits and bifurcation of maps of the interval,, SIAM J. Appl. Math., 35 (1978), 260. doi: 10.1137/0135020.
[31]	A.-A. Yakubu, N. Li, J. M. Conrad and M.-L. Zeeman, Constant proportion harvest policies: Dynamic implications in the Pacific halibut and Atlantic cod fisheries,, Math. Biosci., 232 (2011), 66. doi: 10.1016/j.mbs.2011.04.004.
[32]	T. Yi and X. Zou, Maps dynamics versus dynamics of associated delay reaction-diffusion equation with a Neumann condition,, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 466 (2010), 2955. doi: 10.1098/rspa.2009.0650.

[1]	Pengmiao Hao, Xuechen Wang, Junjie Wei. Global Hopf bifurcation of a population model with stage structure and strong Allee effect. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 973-993. doi: 10.3934/dcdss.2017051
[2]	Jim M. Cushing. The evolutionary dynamics of a population model with a strong Allee effect. Mathematical Biosciences & Engineering, 2015, 12 (4) : 643-660. doi: 10.3934/mbe.2015.12.643
[3]	Dianmo Li, Zhen Zhang, Zufei Ma, Baoyu Xie, Rui Wang. Allee effect and a catastrophe model of population dynamics. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 629-634. doi: 10.3934/dcdsb.2004.4.629
[4]	Bao-Zhu Guo, Li-Ming Cai. A note for the global stability of a delay differential equation of hepatitis B virus infection. Mathematical Biosciences & Engineering, 2011, 8 (3) : 689-694. doi: 10.3934/mbe.2011.8.689
[5]	Anatoli F. Ivanov, Sergei Trofimchuk. Periodic solutions and their stability of a differential-difference equation. Conference Publications, 2009, 2009 (Special) : 385-393. doi: 10.3934/proc.2009.2009.385
[6]	Elena Braverman, Alexandra Rodkina. Stochastic difference equations with the Allee effect. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 5929-5949. doi: 10.3934/dcds.2016060
[7]	Miljana JovanoviĆ, Marija KrstiĆ. Extinction in stochastic predator-prey population model with Allee effect on prey. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2651-2667. doi: 10.3934/dcdsb.2017129
[8]	Tomás Caraballo, Renato Colucci, Luca Guerrini. Bifurcation scenarios in an ordinary differential equation with constant and distributed delay: A case study. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2639-2655. doi: 10.3934/dcdsb.2018268
[9]	Eugen Stumpf. On a delay differential equation arising from a car-following model: Wavefront solutions with constant-speed and their stability. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3317-3340. doi: 10.3934/dcdsb.2017139
[10]	Jan Čermák, Jana Hrabalová. Delay-dependent stability criteria for neutral delay differential and difference equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4577-4588. doi: 10.3934/dcds.2014.34.4577
[11]	Nika Lazaryan, Hassan Sedaghat. Extinction and the Allee effect in an age structured Ricker population model with inter-stage interaction. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 731-747. doi: 10.3934/dcdsb.2018040
[12]	Qizhen Xiao, Binxiang Dai. Heteroclinic bifurcation for a general predator-prey model with Allee effect and state feedback impulsive control strategy. Mathematical Biosciences & Engineering, 2015, 12 (5) : 1065-1081. doi: 10.3934/mbe.2015.12.1065
[13]	Na Min, Mingxin Wang. Hopf bifurcation and steady-state bifurcation for a Leslie-Gower prey-predator model with strong Allee effect in prey. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 1071-1099. doi: 10.3934/dcds.2019045
[14]	Azmy S. Ackleh, Keng Deng. Stability of a delay equation arising from a juvenile-adult model. Mathematical Biosciences & Engineering, 2010, 7 (4) : 729-737. doi: 10.3934/mbe.2010.7.729
[15]	Gang Huang, Yasuhiro Takeuchi, Rinko Miyazaki. Stability conditions for a class of delay differential equations in single species population dynamics. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2451-2464. doi: 10.3934/dcdsb.2012.17.2451
[16]	Pierre Magal. Global stability for differential equations with homogeneous nonlinearity and application to population dynamics. Discrete & Continuous Dynamical Systems - B, 2002, 2 (4) : 541-560. doi: 10.3934/dcdsb.2002.2.541
[17]	Cuilian You, Yangyang Hao. Stability in mean for fuzzy differential equation. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1375-1385. doi: 10.3934/jimo.2018099
[18]	Rui Hu, Yuan Yuan. Stability, bifurcation analysis in a neural network model with delay and diffusion. Conference Publications, 2009, 2009 (Special) : 367-376. doi: 10.3934/proc.2009.2009.367
[19]	P. Dormayer, A. F. Ivanov. Symmetric periodic solutions of a delay differential equation. Conference Publications, 1998, 1998 (Special) : 220-230. doi: 10.3934/proc.1998.1998.220
[20]	Anatoli F. Ivanov, Musa A. Mammadov. Global asymptotic stability in a class of nonlinear differential delay equations. Conference Publications, 2011, 2011 (Special) : 727-736. doi: 10.3934/proc.2011.2011.727

2018 Impact Factor: 1.313

American Institute of Mathematical Sciences